Since suffering a cruciate knee ligament injury in an English Premier League tie against Manchester United on December 4, the former Crystal Palace winger has been sidelined.
And in recent weeks, he has been reported to have stepped up his recovery process with training sessions with the U23s and the Toffees' main setup.
In his first pre-match briefing at the helms of the Goodison Park outfit - and ahead of Sunday's Merseyside derby against Liverpool - Allardyce stated that the 28-year-old could be near a return to action, given his healing process continues, uninterrupted.
"Bolasie has started training with us, which is good news," Allardyce said.
"A couple of games with the Under 23s or a behind closed doors game [could be all that he needs].
"As long as there is no reaction to the knee, we could see him on the subs bench to begin with."
Bolasie's endured an injury-hit debut season at Everton on the back of a big move from Crystal Palace. He was allowed just 13 top-flight appearances with a solitary strike to show forth.